Maybe it's a simple question... Fix a positive $N$. Let $a_{n}$ be the recursive sequence:

$$a_{1} = N$$
$$a_{n}=a_{n-1}-(a_{n-1})^{\frac{2}{3}}.$$

How many steps do I have to complete in order to have $a_{n}\leq 1$?
And, more general, changing $\frac{2}{3}$ by $\frac{2+\varepsilon}{3}$...

The problem here is that I'm not trying to find the limit, but the moment where I achieve the quantity 1. 
This question is related to the more general one: 
Given a recurrence equation, are there techniques for estimating asymptotics of the solution?
And, too, in graph theory, there are processes where $N$ is the number of edges and the steps here would be the amount of components of your graph. Basically, the techniques that could answer these recursive questions would help on graph's counting problem.

Do you have any hints?
Thanks a lot